Complex Fractions

Question

x + 1 - 6

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x

= x^2 + 2 - 6

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x

Can someone please help with the steps to this conclusion?

Answer 1

If your question is to find the common point between two systems then the following would apply

1) Combine like terms where possible

2) Remove the denominator

3) Combine like terms again

4) Solve for x

x + 1 -(6/x) = (x² + 2 - 6)/x

1) x + 1 -(6/x) = (x² - 4)/x

2) Mulitiply both sides of the equation by x to remove the denominator

x((x + 1 -(6/x)) = x(x² - 4)/x

The x multiplies and cancels out where appropriate. On one side it cancels out completely see below

x² + x - 6 = (x² - 4)

3) Combine like terms again as follows: move the variables over to one side of the equation and the numbers to the other side

x² + x - 6 = x² - 4

x² - x² + x = -4 + 6

This leaves

x = 2

You can graph the original equations at Desmos.com or with a calculator to confirm that 2 is a common point or check it both equations.

x + 1 -(6/x) = (x² + 2 - 6)/x

2 + 1 - 6/2 = (2² + 2 - 6)/2

3 -3 = (4 + 2 - 6)/2

0 = (6 - 6)/2

0 = 0/2

0 = 0

They are not equal graphs but they do share a common point.

There is more than one way of finding the common point since at a certain point both sides can be factored.

I'll leave it to you to check the graphs and try other methods of finding the common point.

Answer 2

Hi Brian,

They've found a way to get a common denominator so they could subtract 6/x.

The challenge is to manipulate (x+1) into a fraction with denominator = x.

Start with (x+1)/1 and multiply top and bottom of this fraction by x.

x*(x+1) / (1*x)

= x*(x+1) / x

= (x²+x) / x

So now we know that (x+1) = (x²+x) / x

We can subtract 6/x easily now because they have common denominator of x.

Subtract the numerators and keep the denominator to follow proper fraction rules.

The answer is:

x²+x-6

---------

x

So, it appears there is a mistake in your question.

It is wrong that x+1 - 6/x = (x^2+2-6)/x

You can verify this on a graphing calculator by plugging x+1 - 6/x into Y1 for your graph, and plug (x^2+2-6)/x into Y2 for your graph, and you'll see different graphs.

Plug x+1 - 6/x into Y1 for your graph and (x^2+x-6)/x into Y2 for your graph and you'll see they are the same graph.

Hope this helps! :)